Information consensus in distributed multiple vehicle ...

Pmrredings of the 42nd IEEE

Conkrenee on Decision and Control Maui, Hawaii USA, December 2003

WeMO1-4

Information Consensus in Distributed Multiple Vehicle Coordinated Control Randal W. Beard, Vahram Stepanyan Department of Electrical and Computer Engineering Brigham Young University Provo, UT 84602, USA Vehicle i

Absfracl-Cooperation in multiple vehicle teams requires information to be s h a d between team members. If shared information is not synchronized across the team, then cooperation is adversely affected. This paper considers the problem of information consensus in multiple agent teams. We define notions of asymptotic consensus and show that a team of agents can be asymptotically brought into consensus if and only if the associated communicution topology admits a spanning tree. A linear consensus strategy is proposed and demonstrated vta several simulation examples.

I. INTRODUCTION In recent years there has been significant research in the area of coordinated control of multiple vehicle systems. Examples include spacecraft formation flying [I], [2], UAV formation flying [3], formation control for underwater vehicles [4], coordinated rendezvous of UAVs [5], [6], coordinated path planning 171, task coordination for UAVs 181, [91, and multiple robot coordination [IO], [Ill. Most of the current literature on multiple vehicle cooperative control assumes perfect and unlimited communication between the agents. However, it has been recognized that limited communication and information flow among vehicles, will significantly impact the ability to coordinate action [12]. The current trend in cooperative control is to distribute the decision making among the vehicles [13], [14], [15], [16], [17], 1181. There are several advantages to distributed decision making including enhanced robustness due to the fact that there is no single point of failure, and the ability for dynamic role assignment. However, distributed schemes are usually based on reactive or behavioral methodologies, which are often difficult to direct toward specific desired plans. On the other hand, centralized planning schemes are often more flexible and powerful, enhancing the ability to coordinate action through deliberative planning techniques. An obvious remedy is to design a deliberative, centralized planning scheme, and then instantiate the scheme on each vehicle in a decentralized implementation 181, 161. If each vehicle instantiates the same algorithm with identical input data, then each vehicle will produce the same plan-of-action, and the vehicles will be “coordinated.” However, if the input data on the vehicles differ, then each instantiation of the centralized algorithm will produce a different result, adversely affecting the coordination between vehicles. Therefore, the different vehicles must form a consensus on their shared data. Consensus can take place at either the input or the output of the coordination algorithm as shown in Figure I . Several

0-7803-7924-1/03/$17.00 02003 IEEE

Input data

Consensus

;

i output Data i Consensus

Fig. 1. Consensus can be formed at eiIher the input, or the output of the cooperative planning algorithm on each vehicle.

issues complicate the data consensus problem. First, the communication links between vehicle pairs are unreliable and are established and broken at random time instances. Second, the communication links generally have limited range. Therefore if the separation between vehicles exceeds a certain distance, then communication will be lost. Third, there is limited communication bandwidth, thus limiting the amount of information that can be exchanged. The fourth complication is that at each time instant, the communication topology my not be fully connected which limits the ability of two vehicles which are not directly connected to form consensus on their information. The objective of this paper is to introduce a scheme that enables the information consensus among vehicles in the presence of limited and unreliable communication with timevarying topology. As a first step in this direction we will make the following simplifying assumptions: Shared information is defined over the field of real numbers, Each agent has a single item of information, The information contained on each vehicle is considered to be equally reliable, i.e., information is weighted equally.

.

11. PROBLEM STATEMENT In this section we formally state the problem addressed in the paper. To make our statements precise we will use terminologyfromgraphtheory[19].Letd= { A & 7 1,2, ...,A’} be a set of N agents whose actions are to be coordinated

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in some fashion. We assume that communication between agents can be both unidirectional and bidirectional. Let E be a directed graph with N vertices representing the agents and with edges representing unidirectional communication links between agents. Agent Ai is said to he a neighbor of agent Aj if there is a directed link from A , to A j . A path is a sequence of distinct vertices such that consecutive vertices are neighbors. If there is a path between any two vertices of a graph, then E is said to be connected. If D is connected, then the group of agents is said to he connected. A directed tree is a directed graph, where every vertex, except the root, has exactly one parent. A spanning tree of a directed graph is a tree formed by graph edges that connect all the vertices of the graph.The communication topology at time t will he represented by the adjacency matrix [20] G(t),where

Gij(t) =

1,

(

0,

In this paper we will assume that the information variables are updated according to a continuous update law of the form

ii = fi

.

.

( ~ 1 ,.. ,ZN, G(t))

(1)

Let x = ( X I , ... ,X N ) be the collection of information variables, then Equation (1) can he written as

(2)

Let

We will assume that G ( t ) is piecewise constant in time, hut that changes in the elements of G ( t ) occurs randomly in time. As an example, the directed graph shown in Figure 2, has an adjacency matrix

(‘iB”) 0

111. LINEAR CONSENSUS FILTER

x = f (x,G ( t ) ).

if there is an edge from A j to Ai otherwise

G=

reachable if there exists an information update strategy for each x i , i = 1,.. . ,N that achieves global consensus asymptotically for A. In this paper we provide necessary and sufficient conditions under which A is global consensus reachable, and propose a simple information update strategy that achieves global consensus asymptotically for A under these conditions.

1 0 0 0 .

1 0 - 0 0 0 The double arrows in Figure 2 constitute a spanning tree of the graph.

s = {x E

R - W : x, = x* =

... = x*.},

then the following theorem follows directly from Definition 2.1 7heorem 3.1: The set of agents A is global consensus reachable if and only if S is a positively invariant set of system (2). In the remainder of the paper we will assume that the communication graph 0 is not time-varying, and show by construction that A is global consensus reachable if and only if E has a spanning tree. Toward that end, we propose using the linear update scheme N

*., - ~ o , j G J i ( x j - x ; ) , d = l , 2 , . . N , ~

(3)

j=1

Fig. 2. A directed graph and its spanning tree.

Associated with each agent is an information variable x i , 1 , . . . ,N.To simplify .notation, in this paper we will assume that x i ( t ) E R,and that x i ( t ) are continuously differentiable functions in time. Alink from A, to Ai implies that xj is communicated to Ai. Definition 2.1: The set of agents A is said to he in consensus at time to. if t to implies that Ilzi(t)- xj(t)ll = 0 for each ( i , j ) = 1,.. . ,N. The set of agents A is said to reach global consensus asymprorically if for any x;(O), i = 1 , . ..,,N, Ilzi(t)- xj(t)ll + 0 as t -t M for each (i, j) = 1,. . . ,N.The set A is said to be global consensus

i

=

>

where u;j are positive constants. The essential idea is that if there is a communication link from Aj to Ai (i.e., Gij = l), then agent Ai will update its information variable in the direction of the information variable of agent A,. The constants u,j represent the magnitude of the update and are a function of the relative confidence that agents A , and A j . have that their information variables are correct. For example, if agent A; has much higher confidence that x i is correct then agent Ai’s confidence that zj is correct, then uij will he small, effecting a small movement from xi to xj. On the other hand, if agent A; has much lower confidence that xi is correct then agent Aj’s confidence that xj is correct, then oi3will be large, effecting a large movement from xi to xj. In this paper we will assume that uij is a positive constant that is specified a priori. In future work, we will relax this assumption and allow uij to vary as consensus is formed. Writing the system of equations (3) in matrix form gives =

I

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[

- (E:b;;Gi;)

cizclz -(L,N_*.2jG23)

CNlONl

“NZCNZ

...

‘INGIN

” ’

..

-(E&>

-N;..V~)

To simplify notation let B be a square matrix with B, - gEilGijand let n be a diagonal matrix with rrii = CY=roEilGij. Therefore Equation (4) becomes j,

= ( B - n)x.

(5)

Note that S is invariant under Eq. (5). We need to show that in fact S is positively invariant under Eq. (5). To motivate our main result, we will consider several academic examples. Consider the communication graph shown in Figure 3. If uij = 1 for all z , j , then B - n is of the form

Note that if agent Ai is isolated, then both the &ow and the i'hcolumn of B - will be zero. In this case, B - Il has two zero eigenvalues and two negative eigenvalues. Therefore, in this case S is not positively invariant. Intuitively it is clear that agent A4 cannot form consensus with the (AI, A?, A3) group. Isolated agents are indicated in the structure of B - n by an &ow and column being simultaneously equal to zero. Another case which is not global consensus reachable is shown in Figure 5. In this case we have

n

Fig. 5. Fig. 3. Communication network with four agents,

Graph with two leaden

-11

Note that B - II has one zero eigenvalue (corresponding to S-invariance), and three eigenvalues at -1, -1, and -3. Therefore system ( 5 ) for this example renders S positively invariant. Also note that the graph in this example has a spanning tree with corresponding B - matrix equal to

n

1

0

0

0 - 01 10) '

0

0

0

0

Again B - ll has two zero eigenvalues with the two negative eigenvalues, and the corresponding system does not render S positively invariant. It is intuitively clear that a group with two leaders is not global consensus reachable since the leaders do not have any mechanism to form consensus between themselves. The multiple leader scenario is indicated by more that one zero row in B - ll. As a final example of graphs that are not global consensus reachable, consider the case where G contains two or more connected subgroups that are disconnected from each other, e.g., Figure 6. In this case we have

which has eigenvalues at 0, -1, -1, -1. Therefore adding new edges to a tree does not change the qualitative nature of the eigenvalues, i.e. there is one zero eigenvalue and others in the open left half plane. Consider the graph shown in Figure 4, which has an isolated agent. In this case we have Fig. 6. Communication network with two disconnected subgroups.

Fig. 4. A single isolated agent.

-1

.-n=(g

x ;!). 0

-1

0

-

which has two eigenvalues equal to zero. The following lemma shows that the existence of a spanning tree of 0 is necessary and sufficient for A to be global consensus reachable. Theorem 3.2; The group of agents A is global consensus reachable if and only if the associated communication graph G has a spanning tree.

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Proof: (Sufficiency.)Without loss of generality, re-number the agents such that they are numbers successively according to their depth in the spanning tree, with the mot numbered as agent A l . In other words, children of A1 are numbered A2 to Aql, children of A:! to A,, are labeled A,,+1 to A,, and so on. Let G be the adjacency matrix associated with the spanning tree of 9, and consider the information update scheme N

fi = CGji ( z j - x i ) ,

i = 1,2, ..N.

j=1

Let B = G and bij, giving

E,"=,

fi

be the diagonal matrix with

x=

~

(B-li)x.

+ii

=

(8)

Since the root A1 bas no parents, the first row of B - li will be equal to zero. Since each remgning node has at most one parent, the' diagonal elements of B - II will be equal to -1. The renumbering of agents was performed such $at if A, is a child of A, then q < p which implies that B - fi will be lower triangular. Therefore B - ri as one eigenvalue at 0 and the remaining eigenvalues at -1, which implies that S is positively invariant. (Necessity.) Suppose that A is global consensus reachable but that D does not have a spanning tree. Then there exist at least two agents Ai and Aj such that there is no path in B that contains both Ai and A3. Therefore it is impossible to form consensus between these two agents which implies that A is not global consensus reachable. The proof of Theorem 3.2 was constructive in that equation (8)can be used to form consensus in the group. However, it may be the case that many of the connections are being ignored. The update law in Eq. (3) accounts for all known connections. The issue is whether Eq. (3) causes A to reach global consensus asymptotically when 9 has a spanning tree. The following theorem partially answers the question. Definition 3.3: A graph 9 is said to be of Class CAS if one of the following conditions holds:

Otherwise, using the spanning tree of 9, re-number the nodes as described in the proof of Theorem 3.2. Subsequently, perform the following change of variables Y1 = X I ; yi=x,,

i = 2 ,..., N .

Let T be the associated transformation matrix, where t j l = 1:j = l , . . . ,N and tii = -1,i = 1,.. . , N and zeros otherwise. Letting P = T ( B - II)T-' and y = (yl, .. . ,y ~ we have y = Py. Suppose that case 2 in definition 3.3 holds. Then there is an edge in D that returns to the root of the spanning tree which implies that B - II does not have a zero row. After the state transformation, we can compute the structure of the matrix P as follows. Elements in the first column of matrix P can be calculated as N

N

pi1 = C x t i J b t i

- n7L)tG1

r=1 I=1

N

N

= x t i i ( b i l -aii)tnl + C t i i ( b i i - ail)t&' 1=1

1=1

N

N

= c ( b u - nii)til - x ( b i l l=I

- nir)t;'.

1=1

Recalling the definition of matrix II and that the first column of T-' consists of ones we get for i = 1,... ,N

1=1

(=I

For the other columns of matrix P we calculate as follows:

1) D is a tree, 2) There exists a spanning tree of 9 such that the rmt of the spanning tree is the child of at least one other edge in 0. 3) There exists a spanning tree of 9 such that the root of the spanning tree has at least N - 2 children. Note that the graphs in Figures 2 and 3 are of class CAS, whereas the graphs in Figures 4, 5 , and 6 are not. Theorem 3.4: The group of agents A reaches globally consensus asymptotically using the update law (3) if its associated communication graph 9 is of class CAS. Proof: If 9 is a tree, then the proof is identical to the sufficiency proof of Theorem 3.2.

1=1

l=l

In the j t h column of matrix T-' there is only one nonzero element, namely t;; = -1, therefore for each ( i , j ) = 1,.. . ,N we have pij = -bij +ai3

+ bij - R . -- b. '3

23

- b y - "ij

Case 2 implies that there are no rows of B that equal zeros. We show that in this case the ( N - 1)" order minor

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)

~

P11 of matrix P corresponding to the element p l l is strictly diagonally dominant. Taking into account that xij = 0 if i # j we can write " n

n